Classification of generalized multiresolution analyses (Q973956)

This paper establishes in Theorems 12 and 13 classification of GMRAs (generalized multiresolution analyses), both classical and those defined on abstract Hilbert spaces, by their multiplicity functions \(m\) in Definition 2 and matrix-valued filter functions \(H\) in Definition 4. Moreover, a construction procedure is described in Theorem 8 that produces an abstract GMRA with multiplicity function \(m\) and filter system \(H\). The classification results are then applied to many previously studied classical and abstract examples in Section 5. The interesting results in this paper deepen our understanding of GMRAs, which play the core role in wavelet analysis.